So, watch the video and I'll tell you how I did it, and what I actually wanted to do.
1. Make a cube out of 5 copies of the same plane, made the front face from a different plane - in retrospect, could probably lose a few faces for speed where they're not visible.
2. Added an image layer, made it invisible and added an Environment Map viewer to that front plane and made it use the image layer.
3. Added 16 of the Cube Composite to another Composite and made them all 3D Unrolled (as nothing else worked right)
4. Keyframed the rotations of each cube at a random time and either left/right, or up/down.
5. Added couple of random lights: one above and in front, one behind and to the side.
6. Pointed the camera at one of the cubes in the middle and keyframed two positions for it at the start and end, so it orbited around them all slightly.
This was done in Hitfilm 4 Express, BTW.
It looks OK(ish), but because the Environment Map Viewer is projecting the image onto the face, it's not rotating with it, which is what I wanted.
I'm probably using the wrong tool entirely, but with Hitfilm, you can sometimes find a happy accident that still does something interesting.
There is a line between all the cubes, despite them being 200x200x200 and 200 pixels apart. I guess I'll have to try and make them 199 pixels apart to overlap slightly or make them 199x199x199 so there is an actual centre pixel (odd numbered dimensions will have one) rather than it being off-centre by half a pixel.
That, I can mess about with easily enough, but is there a better way to 'project' the image into the front face of the cube surface so that it rotates with it, short of having to cut the image up and apply it to each cube individually - and therefore have 16 different cubes instead of one? Or not?